Mathematics

Trigonometric Identities and Equations

223 Questions

Solve a variety of questions based on trigonometric identities and mathematical equations. Key areas covered include double angle formulas, the law of sines, and quadrant angles. This material helps students preparing for advanced mathematics exams, UPSC, and state public service commissions.

Double angle formulasLaw of sinesTrigonometric quadrantsLaplace transformSecant functionTaylor series

Trigonometric Identities and Equations Questions

Multiple choice

The trigonometric ratio that relates the length of the opposite side to the length of the hypotenuse is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Secant
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Multiple choice

The trigonometric ratio that relates the length of the opposite side to the length of the adjacent side is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Secant
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Multiple choice

The trigonometric ratio that relates the length of the hypotenuse to the length of the opposite side is called the:

  1. Sine

  2. Cosine

  3. Tangent

  4. Cosecant
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The cosecant of an angle is defined as the ratio of the length of the hypotenuse to the length of the opposite side.

Multiple choice

What is the sine rule?

  1. \(\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}\)
  2. \(\frac{sin A}{a} = \frac{cos B}{b} = \frac{tan C}{c}\)
  3. \(\frac{cos A}{a} = \frac{sin B}{b} = \frac{tan C}{c}\)
  4. \(\frac{cos A}{a} = \frac{cos B}{b} = \frac{sin C}{c}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sine rule states that in a triangle, the ratio of the sine of an angle to the length of the opposite side is the same for all angles.

Multiple choice

What is the cosine rule?

  1. \(c^2 = a^2 + b^2 - 2ab cos C\)
  2. \(c^2 = a^2 + b^2 + 2ab cos C\)
  3. \(c^2 = a^2 - b^2 + 2ab cos C\)
  4. \(c^2 = a^2 - b^2 - 2ab cos C\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cosine rule states that in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle between them.

Multiple choice

What is the double angle formula for sine?

  1. \(sin 2A = 2 sin A cos A\)
  2. \(sin 2A = sin A + sin A\)
  3. \(sin 2A = cos A + cos A\)
  4. \(sin 2A = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double angle formula for sine states that (sin 2A = 2 sin A cos A).

Multiple choice

What is the double angle formula for cosine?

  1. \(cos 2A = cos^2 A - sin^2 A\)
  2. \(cos 2A = cos A + cos A\)
  3. \(cos 2A = sin A + sin A\)
  4. \(cos 2A = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double angle formula for cosine states that (cos 2A = cos^2 A - sin^2 A).

Multiple choice

What is the half angle formula for sine?

  1. \(sin \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{2}}\)
  2. \(sin \frac{A}{2} = sin A + sin A\)
  3. \(sin \frac{A}{2} = cos A + cos A\)
  4. \(sin \frac{A}{2} = 2 sin A cos A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half angle formula for sine states that (sin \frac{A}{2} = \pm \sqrt{\frac{1 - cos A}{2}}).

Multiple choice

What is the half angle formula for cosine?

  1. \(cos \frac{A}{2} = \pm \sqrt{\frac{1 + cos A}{2}}\)
  2. \(cos \frac{A}{2} = cos A + cos A\)
  3. \(cos \frac{A}{2} = sin A + sin A\)
  4. \(cos \frac{A}{2} = 2 cos A sin A\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half angle formula for cosine states that (cos \frac{A}{2} = \pm \sqrt{\frac{1 + cos A}{2}}).

Multiple choice

Find the value of $x$ in the equation $\sin x = \frac{1}{2}$.

  1. $30\degree$
  2. $45\degree$
  3. $60\degree$
  4. $90\degree$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the inverse sine function, we get $x = \sin^{-1}\left(\frac{1}{2}\right) = 30\degree$.

Multiple choice

What is the reciprocal identity for sine?

  1. sin(x) = 1/csc(x)

  2. sin(x) = 1/sec(x)

  3. sin(x) = 1/tan(x)

  4. sin(x) = 1/cot(x)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The reciprocal identity for sine is sin(x) = 1/csc(x).

Multiple choice

What is the double-angle identity for cosine?

  1. cos(2x) = cos^2(x) - sin^2(x)

  2. cos(2x) = 2cos^2(x) - 1

  3. cos(2x) = 1 - 2sin^2(x)

  4. cos(2x) = 2cos(x) - 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The double-angle identity for cosine is cos(2x) = cos^2(x) - sin^2(x).

Multiple choice

Which of the following is the half-angle identity for sine?

  1. sin(x/2) = sqrt((1 - cos(x))/2)

  2. sin(x/2) = sqrt((1 + cos(x))/2)

  3. sin(x/2) = (1 - cos(x))/2

  4. sin(x/2) = (1 + cos(x))/2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The half-angle identity for sine is sin(x/2) = sqrt((1 - cos(x))/2).

Multiple choice

What is the sum-to-product identity for sine and cosine?

  1. sin(x) + cos(x) = 2sin((x+y)/2)cos((x-y)/2)

  2. sin(x) + cos(x) = 2cos((x+y)/2)sin((x-y)/2)

  3. sin(x) + cos(x) = sin((x+y)/2) + cos((x-y)/2)

  4. sin(x) + cos(x) = sin((x+y)/2) - cos((x-y)/2)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum-to-product identity for sine and cosine is sin(x) + cos(x) = 2sin((x+y)/2)cos((x-y)/2).

Multiple choice

Which of the following is the difference-to-product identity for sine and cosine?

  1. sin(x) - cos(x) = 2sin((x-y)/2)cos((x+y)/2)

  2. sin(x) - cos(x) = 2cos((x-y)/2)sin((x+y)/2)

  3. sin(x) - cos(x) = sin((x-y)/2) + cos((x+y)/2)

  4. sin(x) - cos(x) = sin((x-y)/2) - cos((x+y)/2)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The difference-to-product identity for sine and cosine is sin(x) - cos(x) = 2sin((x-y)/2)cos((x+y)/2).